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Calculation of synchronization within cognitively informed framework
This protocol is extracted from research article:
Cognitive chimera states in human brain networks
Sci Adv, Apr 3, 2019;

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We used the standard order parameter ρ to estimate the extent of synchronization after a targeted regional stimulation within the brain networks. This measure was proposed by Kuramoto for the estimation of coherence in a population of Kuramoto phase oscillators (33). In this case, the instantaneous order parameter at a given time t was defined as$ρN(t)eiΦ(t)=1N∑j=1Neiϕj(t)$(4)where ϕj is the phase of the jth oscillator at time t and is given by$ϕj(t)=tan−1Ij(t)Ej(t)$(5)

Here, N = 76 is the number of oscillators in the system. To estimate the global synchronization in the system, one needs to average the instantaneous order parameter for a sufficiently long period of time (T).$ρN=<ρN(t)>T$(6)

We used 1 s of simulated activity to estimate the average order parameter. Within our cognitively informed framework, we measured the synchronization between all pairs of cognitive systems following a regional stimulation. This was performed by calculating an order parameter for the combined oscillator population of a pair of cognitive systems. For cognitive systems si and sj, this order parameter is given by$ρsi,sj=<ρsi,sj(t)>T$(7)where$ρsi,sj(t)eiΘ(t)=1Nsi+Nsj∑k∈(si∪sj)eiϕk(t)$(8)

Here, $Nsi$ and $Nsj$ represent the number of oscillators (brain regions or nodes) within cognitive systems si and sj, respectively. This analysis resulted in synchronization matrices, as shown in Fig. 2A, whose entries represent the extent of synchronization between cognitive systems. These matrices were used to identify the dynamical cognitive state that emerged as a result of regional stimulation.

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